3
votes
1answer
40 views

Trying to prove lim inf $(A_n) \subseteq$ lim sup $(A_n)$

I am trying to show that for a sequence of sets $(A_n)$ $$ lim \ inf \ (A_n) \subseteq lim \ sup \ (A_n)$$ I can see why this is true intuitively as follows - $x \in lim \ inf \ (A_n)$ $\implies ...
4
votes
1answer
90 views
+50

Halmos “Measure theory” exercise on limit of sequence of sets

Problem statement. This is an exercise from chapter 1, section 4 (problem 13) from Halmos textbook: If $\{E_n\}$ is a sequence of sets, write $D_1=E_1, D_2=D_1 \triangle E_2, D_3=D_2 \triangle E_3$ ...
2
votes
1answer
58 views

Set convergence and lim inf and lim sup

I'm a bit confused with the general concept of convergence of a sequence of sets. I'm well aware that the limit of a sequence $\{C^{\nu}\}$ exists iff $$\liminf_{\nu \rightarrow \infty} C^{\nu} = ...
1
vote
2answers
27 views

does taking limsup preserve inclusion relation?

Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of sets with no further structure at that point, such that $a_n \subset b_n$ for every $n\in \mathbb{N}$, does it holds that ...
0
votes
2answers
57 views

Disjoint union with limsup

For any sets $A_n,n\in\mathbb{N}$ consider $$ A^+:=\limsup_{n\to\infty}A_n:=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k,~~~~~E_m:=\bigcup_{n\geq m}A_n. $$ Show that the sets $E_m, ...
0
votes
0answers
35 views

Indicatorfunction of limes superior resp. limes inferior

Consider subsets $A_1,A_2,\ldots$ of a set $X$ and $$ A^+:=\limsup_{n\to\infty}A_n,~~~~~A^-:=\liminf_{n\to\infty}A_n. $$ Show that $$ 1_{A^+}(x)=\limsup 1_{A_n}(x),~~x\in ...
1
vote
1answer
37 views

Show: $\limsup$ does not change when changing finite many sets

Let $(A_n)_{n\in\mathbb{N}}$ be a series of sets. Define $$ ...
2
votes
2answers
127 views

Lim sup of sequence of sets and theirs unions [closed]

I have to prove the following equality: Can somebody help me to prove this?
0
votes
1answer
143 views

Understanding the supremum limit of a set

Given a sequence ${A_n}$, we define the set lim sup $A_n = \{x : x$ belongs to infinitely many $A_n$'s$\}$ That is - lim sup $A_n = \bigcap_{m=1}^\infty (\bigcup_{n=m}^\infty A_n)$ I can't see how ...
1
vote
1answer
262 views

Liminf and Limsup in measure theory and in sequences

In measure theory, given sets $A_1,A_2,\ldots$, we define $\liminf A_n=\bigcup_{k=1}^\infty\left(\bigcap_{n\geq k}A_n\right)$ and $\limsup A_n=\bigcap_{k=1}^\infty\left(\bigcup_{n\geq k}A_n\right)$. ...
2
votes
1answer
146 views

Lim Inf and Lim Sup of Collection of Sets

Folland defines $$\limsup E_n=\bigcap_{k=1}^{\infty}\bigcup_{n=k}^{\infty}E_n,\liminf E_n=\bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}E_n.$$ And states that $\limsup E_n=\{x:x\in E_n$ for infinitely ...
2
votes
1answer
83 views

Is the upper limit of sets empty?

Let $E_k^n$ be measurable subset of $[0,1]$ for any natural numbers $k$ and $n$. For a fixed $n$ we have $$E_1^n\supseteq E_2^n\supseteq E_3^n\supseteq\ldots E_{k}^n\supseteq ...
4
votes
1answer
61 views

Is it true that $\cup_{n=N}^{\infty}A_{n}\setminus\cap_{n=N}^{\infty}A_{n}=\cup_{n=N}^{\infty}A_{n}\triangle A_{n+1}$?

During an exam I have claimed that if $\{A_{n}\}_{n=1}^{\infty}$ then for any $N\in\mathbb{N}$ $$\limsup A_{n}\setminus\liminf ...
0
votes
1answer
849 views

limsup liminf of sequence of sets

Following up from the discussion here: Liminf and Limsup of a sequence of sets I wanted to confirm my understanding of these concepts with another example. Suppose we have: $a_n>0$, $b_n >1$ ...
1
vote
3answers
259 views

A problem in Sigma algebra.

How do I conceptualise this expression : Let {$A_n$}$^{n=\infty}_{n=1}$ belong to sigma algebra $A$. Define, $\limsup\{A_{n}\}=\bigcap_{n=1}^{\infty}\{\bigcup_{m=n}^{\infty}A_{n}\} $ and similarly ...
5
votes
4answers
4k views

lim sup and lim inf of sequence of sets.

I was wondering if someone would be so kind to provide a very simple explanation of lim sup and lim inf of s sequence of sets. For a sequence of subsets $A_n$ of a set $X$, the $\limsup A_n= ...
2
votes
2answers
2k views

Interpretation of {Infinitely Often} = {Almost Always}

I am trying to better understand what it means for a sequence $A_n$ of subsets of a set $S$ to be such that $\bigcap^\infty_{n = 1} \bigcup^\infty_{m = n} A_n = \limsup A_n = \liminf A_n = ...
3
votes
2answers
2k views

liminf and limsup with characteristic (indicator) function

So first let me state my homework problem: Let $X$ be a set, let $\{A_k\}$ be a sequence of subsets of $X$, let $B = \bigcup_{n=1}^{+\infty} \bigcap_{k=n}^{+\infty} A_k$, and let $C = ...
5
votes
1answer
1k views

limsup and liminf of a sequence of points in a set

My ways to define/write limsup and liminf of a sequence of points in a set $X$: They come from what I have understood. If you have other ways of understanding, really appreciate if you can reply ...
6
votes
2answers
2k views

limsup and liminf of a sequence of subsets of a set

I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$. It says there are two different ways to define them, but first gives what is common for ...
1
vote
2answers
681 views

Proof: Limit superior intersection

How to prove $\limsup(\{A_n \cup B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!
0
votes
1answer
990 views

limit superior and limit inferior of the given sequence of sets

A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
6
votes
3answers
3k views

limit inferior and superior for sets vs real numbers

I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot ...